Since the fundamental solution systems of neighborhoods of different regular
singularities are different, and many equivalent fundamental solution systems
can be constructed corresponding to neighborhoods of same regular singularity.
When the Feynman integral is expressed as a linear combination of
hypergeometric functions of corresponding fundamental solution system,
there are obvious differences among the equivalent representations accordingly.
Finding a framework for understanding these rambling results is undoubtedly
important on both theoretical and phenomenological aspects.
The fundamental solution systems of neighborhoods of different regular
singularities can be regarded as analytic continuations of each other.
Thus the analytic expression of Feynman integral can be obtained in the whole parameter space.
Taking the Feynman integral as the linear combinations of hypergeometric functions from
the fundamental solution systems around different regular singularities and embedding
it on the subvariety of Grassmannians, one finds that the dual space of the GKZ-system
satisfied by the Feynman integral in the splitting coordinates determines the integer
lattice matrices, and then write the exponent matrices compatible with the integer
lattice matrices. The geometric representation [1]} of these exponent
matrices can clearly tell us which hypergeometric functions of the fundamental
solution systems of neighborhoods of different regular singularities are
proportional to each other.
